Preheat the oven to 425°F (220°C). Lightly grease a baking sheet or line it with parchment paper.
In a bowl, toss the cooked chicken with 1/4 cup buffalo wing sauce until evenly coated. Add more or less sauce according to your preferred spice level.
In a separate bowl, stir together the ranch dressing and remaining 1/4 cup buffalo wing sauce.
On a lightly floured surface, roll the pizza dough into a long rectangle.
Spread the buffalo ranch sauce evenly over the dough, leaving a small border around the edges.
Scatter the buffalo-coated chicken evenly over the sauce.
Sprinkle the mozzarella and cheddar cheeses over the chicken.
Starting from one long side, carefully roll the dough into a log.
Pinch the ends closed to help keep the filling from escaping during baking.
Place the roll seam-side down on the prepared baking sheet.
Bake for about 30 minutes, or until the crust is golden brown and the cheese is melted and bubbling.
Remove from the oven and let the Buffalo Chicken Bake rest for 5–10 minutes before slicing and serving.